1On a Misguided Argument for the Necessity of Identity
Ari Maunu
University of Turku
Department of Philosophy
FI-20014 TURKU
FINLAND
arimaunu@utu.fi
http://users.utu.fi/arimaunu/
Orcid 0000-0003-0259-5493
Abstract
There is a certain popular argument, deriving from Ruth Barcan and Saul Kripke, from the
conjunction of the Principle of the Indiscernibility of Identicals (PInI, for short) and the
Principle of the Necessity of Self-Identity to the Thesis of the Necessity of Identity. My
purpose is to show that this argument does not work, not at least in the form it is often
presented. I also give a correct formulation of the argument and point out that PInI is not even
needed in the argument for the necessity of identity.
Keywords: necessity, identity, rigidity, substitutivity
2I. INTRODUCTION
According to a popular argument for the necessity of identity, deriving from Barcan
1947, Marcus 1961,1 and Kripke 1971, actual identity of an object a with an object b entails
necessary identity of a with b, because on the assumption that a=b the necessity of b’s being
identical with a follows from the self-evident necessity of a’s being identical with a by the
Principle of the Indiscernibility of Identicals (aka, “Leibniz’s Law”), or “whatever is true of a
thing is true of anything identical with that thing” (PInI, for short).
After presenting in Section II E. J. Lowe’s (2002) version of the argument for the
necessity of identity (which I shall call “the putative argument”), I show in Section III that
the reasoning represented by his formulation is mistaken (as an attempt to prove the necessity
of identity). In Section IV I present the correct version of the argument in question. Finally,
in Section V I point out that PInI is not even needed in a conclusive argument for the
necessity of identity.
II. THE PUTATIVE ARGUMENT FOR THE NECESSITY OF IDENTITY
Taking E. J. Lowe’s (2002: 84–90)2 presentation of the argument for the necessity of
identity as an example, Lowe gives the following formulation of this argument (Lowe 2002:
85):
Suppose [...] that a is identical with b, where a and b are any objects whatever. Now, by
the principle of necessity of self-identity, we can say that a is necessarily identical with
a. From this it follows that it is true of a that it is necessarily identical with a. But from
this and the assumption that a is identical with b it follows, by Leibniz’s Law, that it is
also true of b that it is necessarily identical with a – from which it follows that a is
3necessarily identical with b. What we seem to have proved, then, is that if it is true that
a is identical with b, then it is necessarily true that a is identical with b – and
consequently that there cannot be an identity proposition that is merely contingently
true (true in some possible worlds but not in others).
Lowe’s more careful formulation of this argument is as follows (Lowe 2002: 85–86):
(1) For any object x, it is necessarily the case that x is identical with x. [the necessity of
self-identity]
(2) For any objects x and y, if x is identical with y, then whatever is true of x is also true
of y. [Leibniz’s Law]
(3) a is identical with b. [assumption]
(4) It is necessarily the case that a is identical with a. [from (1)]
(5) It is true of a that it is necessarily identical with a. [from (4)]
(6) If a is identical with b, then whatever is true of a is also true of b. [from (2)]
(7) Whatever is true of a is also true of b. [from (3) and (6)]
(8) It is true of b that it is necessarily identical with a. [from (5) and (7)]
(9) It is necessarily the case that a is identical with b. [from (8)]
Therefore,
(10) If a is identical with b, then it is necessarily the case that a is identical with b.
[from (3) & (9)]
Let us utilize lambda abstraction for highlighting the locution ‘is true of’: “λx(Fx)b”, or “It is
true of b that it is an F”, for instance, is equivalent with “∃x(b=x & Fx)”, but not, in general,
with “Fb”, because the position of ‘x’ in “Fx” may be intensional – for example, it may be
4true of b that it is believed by c to be a G (or, λx(BcGx)b), without it being true that c believes
that b is a G (or, BcGb). On the face of it, Lowe’s argument may be formalized as follows:
1* ∀x□x=x
2* ∀xy(x=y → ∀X(λz(Xz)x ↔ λz(Xz)y))
3* a=b
4* □a=a
5* λz(□a=z)a
6* a=b → ∀X(λz(Xz)a ↔ λz(Xz)b)
7* ∀X(λz(Xz)a ↔ λz(Xz)b)
8* λz(□a=z)b
9* □a=b
10* a=b → □a=b
III. THE MISTAKE IN THE PUTATIVE ARGUMENT
 Prima facie, the problem with Lowe’s argument is that, apparently, the step from 4* to 5*
works only if ‘a’ is rigid – it is not true of the shortest spy that he or she is necessarily the
shortest spy, or ~λz(□s=z)s (even though it is true that □s=s) – and the step from 8* to 9*
works only if ‘b’ is rigid – it is not true that, necessarily, the number of apostles is twelve, or
~□t=n (even though it is true that λz(□t=z)n). But positing that ‘a’ and ‘b’ are rigid trivializes
the argument because then the truth of the conclusion 10* is immediate (modulo the existence
of a (i.e., b) in various possible worlds – though if we assume “Kaplan rigidity”, or direct
referentiality, this existence proviso is not needed).
Lowe (2002: 89) notices this threat of trivialization. He says that he does not assume
that ‘a’ and ‘b’ are rigid (because that would mean, according to Lowe (ibid.), that “there is
5no possible world in which ‘a’ and ‘b’ designate different objects”) but only that they are
“purely referential devices” (PRDs for short), which “serve purely to refer to certain objects
and contribute nothing else to the proposition expressed with their aid” (Lowe 2002: 89; cf.
Quine 1961: 140). In particular, definite descriptions are not allowed, for “genuine identity
propositions” cannot, according to Lowe, even be formed by their means (Lowe 2002: 88).
However, PRDs – contributing, by definition, nothing but reference to the proposition –
are directly referential terms (DRTs), for the standard characterization of the latter is that
their semantic function is only to pick out the referent, or refer without a mediation of the
content. And any DRT is rigid in a sense that may be clarified by means of Carnapian
intensions, or (possibly incomplete) functions from possible worlds to entities (individuals,
sets, or truth-values). The purpose of such intensions is to reflect the semantic roles of
expressions. For a singular term ‘b’ in general, the Carnapian intension I (in the domain A of
individuals) is a function from the set of possible worlds, W, to individuals (I(‘b’): W→A),
and for a rigid singular term ‘c’ it is a constant function (I(‘c’): W→{c}). In contrast, for a
singular term ‘d’ whose semantic function “is only to pick out the referent” (i.e., PRDs in
Lowe’s terminology), the Carnapian intension – indicating the difference from rigidity – is
not properly a function at all but reduces to singling out an individual (I(‘d’) = d).3 Although
a PRD may fail to be rigid in the strict technical sense of referring to the same individual in
every possible world, we may still say that a PRD picks out the same individual with respect
to (rather than in) every possible world, or that it is “Kaplan rigid”. All in all, if ‘a’ and ‘b’
are PRDs, and hence DRTs, on a=b there is no possible world with respect to which ‘a’ and
‘b’ designate different objects. This trivializes Lowe’s argument.
Also, if ‘a’ and ‘b’ are PRDs – contributing to the proposition nothing else but the
referent – then if ‘a’ and ‘b’ are co-referring, there cannot be any difference in the
propositions that a=a and that a=b, which means that they are the same proposition (cf. the
6next paragraph). Then, because it is true of that a=a that it is necessary – λx(□x)p, where p is
that a=a – this must, by PInI, be true of that a=b as well: λx(□x)q, where q is that a=b. This,
again, trivializes the argument (for PInI is, of course, trivially true).
In addition to Lowe’s argument being trivial, his resorting to PRDs has the following
embarrassing consequence. Lowe seems to support the celebrated thesis that some necessarily
true propositions, e.g. just “genuine identity propositions”, are a posteriori. Indeed, he states
that “if the argument [(1)–(10)] is correct, it shows us that there can be necessary truths that
are not knowable a priori” (Lowe 2002: 86). However, as just indicated, if a=b, then that a=a
is the same proposition as that a=b, for PRDs ‘a’ and ‘b’. Thus, if it is true of that a=a that it
is a priori, this must, by PInI, be true of that a=b as well.4 (Or perhaps both are a posteriori
because the latter is – how are we to decide?) So, it seems that the proposition that a=b is not
a posteriori after all, a result that is unwelcome to almost everyone (though it has been argued
for by some hardline direct reference theorists, for example Nathan Salmon (1986: 137–38)).
Lowe’s mistake has to do with his misguided attention to signs. If we are trying to
prove the necessity of identity of the object x with the object y, it cannot matter by which
means x (= y) is picked out – thus, definite descriptions, for example, should do as well. That
is, it should not matter whether ‘a’ and ‘b’, as used in the proper argument for the necessity
of identity, are rigid designators (e.g. proper names or indexicals) or nonrigid designators
(e.g. definite descriptions). If the largest planet in the solar system (theL, for short) is
identical with the planet which is fifth nearest to the sun (theF), then it is true of the planet
that is in fact theL (= Jupiter) and the planet that is in fact theF (= Jupiter) that, necessarily,
the former is identical with the latter.5 (By the same token, if the referent (or denotation or
whatever term one wants to use) of ‘theL’ is the same as the referent of ‘theF’, then,
necessarily, the former is identical with the latter.) The identity involved in the true “TheL is
theF” is, like any identity, necessary identity, even though “TheL is theF” does not express a
7necessary truth – there are possible worlds with respect to which “The largest planet is
identical with the fifth nearest” is false, even though (to repeat) the objects that in fact are the
largest planet and the fifth nearest planet are necessarily identical. It is just as Kripke (1980,
3) says:
If ‘a’ and ‘b’ are rigid designators, it follows that ‘a=b’, if true, is a necessary truth. If
‘a’ and ‘b’ are not rigid designators, no such conclusion follows about the statement
‘a=b’ (though the objects designated by ‘a’ and ‘b’ will be necessarily identical).
It might even be said that precisely the fact that definite descriptions won’t do in the putative
argument shows that that argument, as Lowe presents it, cannot be used as an argument for
the necessity of identity, and that this is a clear indication that those who formulate the
argument in question in the manner Lowe does – as does, for instance, Bob Hale (2004: 365–
67) – fail to understand it fully. One should not confuse the necessity of identity with the
question of the necessary truth of identity statements.
Arthur Smullyan (1948) utilized the Russellian analysis of identity statements with
definite descriptions as a way to dispel Barcan’s (1947) original version of (something like)
the putative argument. Such a move is, I gather, behind the claim, made by Lowe (2002: 88)
and others, that identity statements with definite descriptions are not “genuine identity
statements”. However, such statements are, arguably, entirely legitimate identity statements
(and Kripke seems to agree in the passage quoted above) simply because it is utterly natural
to say things like “The largest planet in the solar system is the same as the planet which is
fifth nearest to the sun” and infer from this and the fact that the volume of the largest planet is
1321 times the volume of the Earth that this is a fact about the fifth nearest planet as well. (To
my mind, Lowe’s claim, already quoted above, that “there cannot be an identity proposition
8that is merely contingently true” is misleading or at least unduly incautious.) In any case, we
shall see shortly that at least as far as the argument for the necessity of identity is concerned,
there is no need to resort to the “Smullyan Move” nor to quarrel over the correct “definition”
of ‘identity statement’.
IV. THE NECESSITY OF IDENTITY
Let ‘a’ and ‘b’ be singular terms (any devices that pick out individuals), e.g., definite
descriptions, proper names, or indexicals (as used in definite contexts). The true argument for
the necessity of identity, independent of the nature of the singular terms used, goes as
follows:
1' λxy(□x=y)aa necessity of self-identity
2' a=b → ∀X(λx(Xx)a ↔ λx(Xx)b) PInI
3' a=b assumption
4' λxy(□x=y)aa → λxy(□x=y)ab 3' & 2' (for λx(Xx) = λx(λy(□x=y)a))
5' λxy(□x=y)ab 1' & 4'
6' a=b → λxy(□x=y)ab 3' & 5', conclusion
The conclusion is that if a is identical with b then it is necessary that the former is identical
with the latter. It does not matter whether ‘a’ and ‘b’, are rigid or nonrigid (or “purely
referential” or not). For example, if the capital of Russia is the same as the most populous
city in Europe, then it is true of the city that is in fact the capital of Russia and the city that is
in fact the most populous city in Europe that, necessarily, the former (= Moscow) is identical
with the latter (= Moscow). What is crucial for the conclusiveness of this proof of the
necessity of identity (outdated Quinean qualms about the intelligibility of de re constructions
9notwithstanding), is that ‘a’ and ‘b’ appear only in extensional positions – this means that the
proof cannot be contested by appealing to intensionality.
V. THE ARGUMENT FROM SUBSTITUTIVITY
The general validity of PInI is sometimes doubted, usually on the basis that it
(allegedly) fails for intensional statements (see Jacquette 2011 for a recent expression of the
view that PInI is not universally applicable). It seems clear that this is a mistake: for example,
if a=b and it is true of a that it is believed by c to have the property F (or, λx(BcFx)a), it is,
obviously, true of b that it is believed by c to have the property F (or, λx(BcFx)b).
Anyway, it can be seen from the argument given in the previous section that PInI is not
even needed in the argument for the necessity of identity: it suffices to appeal to the hardly
contestable Principle of Substitutivity in Extensional Positions (PSE): “If a=b (so that ‘a’ and
‘b’ are co-referring) and ‘a’ occurs in a sentence in an extensional position, then ‘a’ may be
replaced truth-preservingly by ‘b’ in that position in that sentence.”
The argument from substitutivity to the necessity of identity is short:
 i. a=b assumption
 ii. λxy(□x=y)aa necessity of self-identity
 iii. λxy(□x=y)ab i & ii, by PSE
 iv. a=b → λxy(□x=y)ab i & iii, conclusion
For example, because the largest planet (= theL) is identical with the fifth planet (= theF) and
it is true of the largest planet and the largest planet that necessarily, the former is identical
with the latter, or λxy(□x=y)(theL)(theL),6 the latter ‘theL’ here, occurring in an extensional
10
position, may by PSE be replaced with ‘theF’, to yield the desired λxy(□x=y)(theL)(theF), or,
“It is true of the largest planet and the fifth planet that, necessarily, the former is identical
with the latter” (note again that I am not claiming that necessarily, the largest planet is
identical with the fifth planet, or □(theL = theF)).7
ENDNOTES
1 Of course, Barcan = Marcus (and even necessarily so).
2 Lowe 2002 is a textbook, primarily aimed at students. So, is it an inappropriate object of
criticism? No – to my mind it is particularly important to get things right in a textbook in
order not to mislead students. In any case, similar mistakes appear in Lowe 2005: 300–4, and
Lowe 2013: 128–33.
3 Cf. Kaplan 1989: 493, 497: “[A DRT is] an expression whose semantical rules provide
directly that the referent in all possible circumstances [that is, possible worlds] is fixed to be
the actual referent. [...] The referent, in a circumstance, of [... a DRT] is simply independent
of the circumstance and is [not] a function (constant or otherwise) of circumstance”
(emphases removed). See also Maunu 2002: 55–60.
4 That is, λx(Rx)q follows by PInI from p=q and λx(Rx)p, where p is the proposition that a=a
and q the proposition that a=b and ‘R’ stands for apriority. I think I need to emphasize here
that ‘p’ is in λx(Rx)p in an extensional (de re) position, which means that the inference to
λx(Rx)q is justified (given that p=q). There cannot be any threat of intensional fallacy,
because the inference is not from Rp (where ‘p’ is in an intensional position) to Rq.
Some might protest that intensionality is not eliminated here because propositions
themselves may be construed as intensional objects, or (I take it), as representable by (or as
just being, according to some) functions from possible worlds to truth values (i.e., as
11
Carnapian intensions of declarative sentences). However, the argument stands even on this
conception of propositions: If the function f is the same as the function g, then λx(Rx)f – ‘f’
not in an intensional position! – entails λx(Rx)g, by PInI. (This issue was raised by an
anonymous referee of Journal of Philosophical Research.)
5 Again, this is theL = theF → λxy(□x=y)(theL)(theF). Definite descriptions ‘theL’ and ‘theF’
are in extensional positions in the consequent. The consequent is not the false □(theL = theF).
6 The principle of the necessity of self-identity has sometimes been doubted on the basis that
it is about self-identity of x rather than about the identity of x with x (see Lowe 2002: 86–87,
Lowe 1982, McKay 1986). However, even though it is difficult to give a colloquial
expression of “λxy(□x=y)aa” (perhaps somebody could be absent-minded enough to say,
“The object a and the object a are such that necessary identity holds between them”), this
cannot be held against the appropriateness of this statement: the unnaturalness of a statement
when expressed in natural language has nothing to do with the correctness of the logical form
of the statement.
7 I thank the anonymous referees of Journal of Philosophical Research for useful suggestions
that improved this paper.
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